Bifurcation Theory : Starting from the time I was a Ph-D student, my interest is with the dynamics that arise after perurbation of degenerate dynamical systems. In particular, I have studied the case where a given vector field possesses a degenerate homoclinic orbit, See Ref [1,2,3,4,5,10]. We show the existence of chaos, strange attractors [1, 3] and the existence of infinitely many homoclinic doubling cascades [2,4]. Techniques developed in [1,3] and those in [7,8] (see section ('normal for theory' below) are used to investigate strange attractor with a large entropy . This later results reveals the only knowledge of low order terms in asymptotics are not always sufficient to describe/anticipate the dynamics.
Mathematics in Biology: Classical normal form theory is used to investigated the dynamics of Predator-Prey systems with their bifurcation pattern. A 2-dimensional predator-prey model with five parameters is investigated, adapted from the Volterra-Lotka system by a non-monotonic response function. A description of the various domains of structural stability and their bifurcations is given. The bifurcation structure is reduced to four organising centres of codimension 3. Research is initiated on time-periodic perturbations by several examples of strange attractors. See [12,13]. In order to classify all possible Morse-Smale portrait, we proceed to a surgery
applied to limit-cycle and elaborate a classification of the Morse-Smale portraits 'Modulo' the limit cycles. This is of great help to anticipate the dynamics and their bifurcation.
Normal Form Theory
: Motivated by the research done in homoclinic bifurcations, we are interested in the Dulac map of germ of a vector field. A result is proposed in  int he case of a saddle in the three-dimensional space. Other result concern a direct (asymptotic) computation of a linearisation, conjugating the germ
with its linear part. First results are obtained in the case of a single vector field [7,8] and later in terms of family . These results ohold in any dimension. A non mooth normal form theory is develop (which generalise that of Poincare-Dulac) that eliminate resonant term by resonant term. This non smooth normal form carries Dulac or compensator expansion. A Dulac expansion is formed by monomial terms that may contain a specific logarithmic factor. In compensator expansions this logarithmic factor is deformed. In  such expansions are studied in the frame of quasi periodic bifurcation. We study analytic properties of compensator and Dulac expansions in a single variable. We first consider Dulac expansions when the power of the logarithm is either 0 or 1. Here we construct an explicit exponential scaling in the space of coefficients, which in an exponentially narrow horn, up to rescaling and division, leads to a polynomial expansion. A similar result holds for the compensator case.